Questions: Find an approximate irrational solution to (3^x=5).

Transcript text: Find an approximate irrational solution to $3^{x}=5$.

Solution

Solution Steps

Step 1: Define the Problem

We need to find an approximate irrational solution to the equation $3^x = 5$.

Step 2: Set Up the Bisection Method

We use the bisection method to find the root of the function $f(x) = 3^x - 5$ within the interval $[1, 2]$ with a tolerance of $1 \times 10^{-6}$.

Step 3: Apply the Bisection Method

The bisection method iteratively halves the interval $[a, b]$ where the function changes sign until the interval is sufficiently small. The midpoint of the final interval is taken as the approximate solution.

Step 4: Obtain the Approximate Solution

After applying the bisection method, the approximate solution is found to be $x \approx 1.4650$.

Final Answer

The approximate solution to the equation $3^x = 5$ is: \[ \boxed{x \approx 1.4650} \]

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